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Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex cone $\mathcal{F}_+^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$ are…
To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of…
We show that every (possibly unbounded) convex polygon $P$ in $R^2$ with $m$ edges can be represented by inequalities $p_1 \ge 0,...,p_n \ge 0,$ where the $p_i$'s are products of at most $k$ affine functions each vanishing on an edge of $P$…
We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or…
For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls), describe their structure and give a formulas fo the volume of the biggest of these domains. This paper is an attempt to…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…
It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…
In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it…
This paper presents an algorithmic study of a class of covering mixed-integer linear programming problems which encompasses classic cover problems, including multidimensional knapsack, facility location and supplier selection problems. We…
The problem of finding the minimizer of a sum of convex functions is central to the field of distributed optimization. Thus, it is of interest to understand how that minimizer is related to the properties of the individual functions in the…
We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and…
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ vertices are not known when $s \ge 4$. In this paper, we construct a family of convex small $n$-gons, $n=2^s$…
We consider linear problems in the worst case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the values of the operator uniformly on a convex and balanced set by means of…
An optimization algorithm for nonsmooth nonconvex constrained optimization problems with upper-C2 objective functions is proposed and analyzed. Upper-C2 is a weakly concave property that exists in difference of convex (DC) functions and…
We view a conic optimization problem that has a unique solution as a map from its data to its solution. If sufficient regularity conditions hold at a solution point, namely that the implicit function theorem applies to the normalized…
The paper considers the minimization of a separable convex function subject to linear ascending constraints. The problem arises as the core optimization in several resource allocation scenarios, and is a special case of an optimization of a…
In this paper, we obtain some new upper bounds for differantiable mappings whose q-th powers are geometrically convex and monotonically decreasing by using the H\"older inequality, Power mean inequality and properties of modulus.
We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak in the case of quadratic map. We…
We provide theory for the computation of convex envelopes of non-convex functionals including an l2-term, and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at…
In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization…